BUD (and then MA onward) for the shortest non-AA flight, compared to MAD and LHR.
But my personal choice (especially if in F) would be BA on LHR. You'll have two short flights (6hrs each) but BA F >> the alternative. And BA has the only real F product of your OW choices (AA, IB, MA).
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BA offers First (though I don't know what they offer LHR-TLV); Iberia offers Business Plus transatlantic and then Business Club (kind of like newer version of AA old Business seats) on to TLV (no F ticket on IB).
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Have you considered flying J on RJ JFK-AMM-TLV?
To me I would greatly prefer one long flight versus two mid-length flights. This particularly applies for red-eye, and seems more comfortable then adding a stop right in the middle of your journey just to get F service.
Otherwise, I think BA JFK-LHR-TLV is your only F option.
I've tinkered around a bit. I think I will buy a oneworld explorer and go on to HKG from TLV. Looks like the 763 service on AA to budapest and malev connection offers the best sleep.
Did you look at the connecting times at BUD? IIRC it's something like a 14-hour layover most days, making you arrive at TLV at oh-dark-thirty. Also remember no Plat/EXP status bonus on MA flights.
If you have the clock time available for that sort of long connection, I might suggest you look at the morning AA or BA departure from JFK to LHR, sleep in a real bed at LHR, then take the morning BA flight to TLV. You won't arrive in Israel that much later than if you'd spent the day wandering around Budapest, but you may feel more human for having slept in a real bed in London, rather than on a faux airline "bed." In our experience the daytime TATL eastbound flights really whack jetlag.